How to List All Possible Outcomes When Two Dice Are Thrown
Probability is also known as a possibility of an event occurring. The probability is never negative and is never more than 1. Some of the real-life situations where probability is used are throwing dice, tossing coins, picking out students from a class, and many more. Probability has long been used in mathematics to approximate how likely events are to occur. Essentially, the probability is the degree to which something is predicted to occur.
What is Dice?
Dice is a tiny block with one to six marks or tints on its boundary that is used in games to generate a random number. Dice are tiny, tossable blocks having a visible border that can stop in the figures shown.
When thrown or rolled, the die comes to a halt and displays a random number from one to six on its upper side, with the occurrence of each event being equally likely. The dice drawn are used for playing board games as a fun way to relax with family and friends.
Possible Outcomes in a Dice
What is Sample Space?
A sample space is a collection of possible outcomes from a random experiment. The sign "S" represents the sample space. The events of an experiment are a subset of the possible outcomes. A sample region may have a range of results depending on the investigation. It is termed a discrete or finite sample space if it has a finite number of outcomes. Curly brackets "{ }" contain the sample spaces for a random experiment.
Different Scenarios to Calculate Dice Probabilities
One dice is thrown- The likelihood of a certain integer happening with one dice is the simplest and most straightforward situation of dice probabilities. Dice shows six possible outcomes. So, the result obtained will be given as: \[P\left( A \right) = \dfrac{\text{no of the outcome of A}}{\text{no of total outcomes}}\]
Two dice are thrown - The likelihood of receiving two 6s by tossing two dice is a rare occurrence as the outcome of one die is independent of the outcome of the other dice. The rule of probability applied in such situations states that separate probabilities must be multiplied together to achieve the outcome. As a result, the formula for this is,
Probability of both \[ = \] probability of first \[ \times \] probability of the second
Questions like two or more than two dice are thrown simultaneously to find the probability of getting a number can be done using the above-mentioned formula.
The total number from two or more dice - If one wants to know the possibility of receiving a specific sum obtained by rolling two or more dice, one must use the basic rule of probability which is
Probability = the number of desired results divided by the total number of results.
Sample Questions
1. A dice has how many faces or sides?
a. 4
b. 5
c. 6
d. 8
Ans: 6
Explanation: A dice is a cuboid that has 6 faces or sides in it.
2. How many possible outcomes would be there if two dice are thrown?
a. 6
b. 12
c. 36
d. 2
Ans: 36
Explanation: One dice has 6 possible outcomes. So, if two dice are rolled then we need to multiply 6 two times which will result in 36. The total number of outcomes in a simultaneous throw of two dice will be 36.
3. Possible outcomes that will come in an experiment are called
a. Sample space
b. Probability
c. Possibility
d. Luck
Ans: Sample space
Conclusion
Dice is a six-faced three-dimensional object which is used to play board games. When a dice is thrown there are different probabilities of getting a particular result which can be calculated by a probability formula. Sample space is all the possible outcomes that we can get in a particular situation and is useful in finding out the probability of large and complex sample space.
FAQs on Sample Space of Two Dice Explained with Examples
1. What is the sample space for rolling two dice as per the CBSE curriculum?
The sample space for an experiment is the set of all possible outcomes. When two standard six-sided dice are rolled, each die has 6 possible outcomes {1, 2, 3, 4, 5, 6}. To find the sample space for two dice, we consider all possible ordered pairs, where the first number represents the outcome of the first die and the second number represents the outcome of the second die. This results in a total of 36 possible outcomes: S = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}.
2. How do you calculate the total number of outcomes in the sample space when rolling two dice?
To calculate the total number of outcomes, we use the multiplication principle of counting. Since the two dice are independent events, the total number of outcomes is the product of the number of outcomes for each die.
- Number of outcomes for the first die = 6
- Number of outcomes for the second die = 6
3. Why are the outcomes (1, 2) and (2, 1) considered different in the sample space of two dice?
The outcomes (1, 2) and (2, 1) are considered distinct because they represent different events. In probability, we often distinguish between the dice (e.g., a red die and a blue die, or the first roll and the second roll). The outcome (1, 2) means the first die shows a '1' and the second die shows a '2'. In contrast, the outcome (2, 1) means the first die shows a '2' and the second die shows a '1'. Although the sum is the same (3), the individual results from each die are different, making them two separate, equally likely outcomes in the sample space.
4. What is the general formula to find the size of the sample space for rolling 'n' dice?
The general formula to find the size of the sample space for rolling 'n' independent, six-sided dice is 6n. Here, '6' represents the number of possible outcomes for a single die, and 'n' is the number of dice being rolled. For example:
- For 1 die, the sample space size is 61 = 6.
- For 2 dice, the sample space size is 62 = 36.
- For 3 dice, the sample space size is 63 = 216.
5. How does the sample space for rolling two dice differ from the sample space for tossing two coins?
The primary difference lies in the number of outcomes for each individual event. A coin has two possible outcomes {Heads, Tails}, while a die has six possible outcomes {1, 2, 3, 4, 5, 6}.
For two coins, the sample space is {HH, HT, TH, TT}, with a total of 2 × 2 = 4 outcomes.
For two dice, the sample space consists of 6 × 6 = 36 ordered pairs. The underlying principle of creating ordered pairs of outcomes is the same, but the size and elements of the sample space are significantly different due to the different number of possibilities in the base experiment.
6. What is an example of an event within the sample space of two dice?
An event is any subset of the sample space. For the experiment of rolling two dice, a common example of an event is 'getting a sum of 7'. To find the outcomes for this event, we look for all pairs in the sample space that add up to 7. These outcomes are: {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}. This set of 6 outcomes is the event E, which is a subset of the total 36 outcomes.
7. In what real-world applications is the concept of a sample space, like that for two dice, useful?
The concept of a sample space is fundamental to probability and statistics and has wide applications beyond games. Examples include:
- Genetics: Predicting the possible genetic combinations (sample space) for offspring based on parental genes.
- Risk Management: Insurers and financial analysts determine the sample space of possible market movements or claims to assess risk.
- Quality Control: A manufacturer might define a sample space of all possible product defects to calculate the probability of a faulty item.
- Cryptography: The set of all possible keys for a cipher forms its sample space, which helps in determining its security strength.
